Permutation and Combination

Permutation
: Permutation means arrangement of
things. The word arrangement is used, if the
order of things is considered.

Combination: Combination
means selection of things. The word selection
is used, when the order of things has no importance.

Example:
Suppose we have to form a number of consisting of three
digits using the digits 1,2,3,4, To form
this number the digits have to be arranged.
Different numbers will get formed depending upon the
order in which we arrange the digits. This is an example
of Permutation.

Now
suppose that we have to make a team of 11 players out of
20 players, This is an example of combination,
because the order of players in the team will not result
in a change in the team. No matter in which order we list
out the players the team will remain the same! For a
different team to be formed at least one player will have
to be changed.

Now let us look at two fundamental
principles of counting:

Addition
rule : If an
experiment can be performed in n ways, &
another experiment can be performed in m ways
then either of the two experiments can be performed in
(m+n) ways.This rule can be
extended to any finite number of experiments.

Example:
Suppose there are 3 doors in a room, 2 on one side and 1
on other side. A man want to go out from the room.
Obviously he has 3 options for it. He can
come out by door A or door B or
door C.

Multiplication
Rule : If a
work can be done in m ways, another work can be done in
n ways, then both of the operations can be
performed in m x n ways. It can be extended to any finite
number of operations.

Example.:
Suppose a man wants to cross-out
a room, which has 2 doors on one side and 1 door on other
site. He has 2 x 1 = 2 ways for it.

Factorial
n : The product
of first n natural numbers is denoted by n!.

n! = n(n-1) (n-2)
..3.2.1.

Ex. 5! = 5 x 4 x
3 x 2 x 1 =120

Note
0! = 1

Proof n! =n,
(n-1)!

Or
(n-1)! = [n x (n-1)!]/n = n! /n

Putting n = 1,
we have

O! = 1!/1

or 0
= 1

Permutation

Number of permutations of
n different things taken r at a
time is given by:-

nPr
=
n!/(n-r)!

Proof:
Say we have n different things a1,
a2 , an.

Clearly the first place can
be filled up in n ways. Number of things left
after filling-up the first place = n-1

So the second-place can be
filled-up in (n-1) ways. Now number of things left after
filling-up the first and second places = n - 2

Now the third place can be
filled-up in (n-2) ways.

Thus number of ways of
filling-up first-place = n

Number of ways of filling-up
second-place = n-1

Number of ways of filling-up
third-place = n-2

Number of ways of filling-up
r-th place = n  (r-1) = n-r+1

By multiplication 
rule of counting, total no. of ways of filling up, first,
second -- rth-place together :-